Estimating the DJI Series by Multifractional Brownian Motion

Abstract

We estimate the stock market and its price dynamics with the multifractional Brownian motion. In our analysis, we use the dataset of the Dow Jones Industrial Average (DJI) time series from March 2009 to June 2015. First, we briefly introduce the definitions and properties of the Brownian motion (Bm), fractional Brownian motion (fBm) and multifractional Brownian motion (mBm). (Ayache and Levy Vehel, 2004). Then we model price processes as exponential of the sum of a regular process and a stochastic process and estimate the Holder exponent. In this paper, we show how a stochastic process like mBm can be applied to simultaneously capture the fluctuations of the asset price dynamics and the long-range dependence of financial time series. Thus, we argue that with a proper functional parameter H(t) we can generate mBm that can reproduce the stylized facts that characterize financial time series.

Keywords: Brownian Motion(Bm), Multifractional Brownian motion (mBm), Hurst Parameter Estimation, Financial Time Series, Asset price dynamics

JEL Classification: C15, C40, G00, G40